Closed linear spaces consisting of strongly norm attaining Lipschitz functionals

Abstract

Given a pointed metric space M, we study when there exist n-dimensional linear subspaces of $$\mathrm {Lip}_0(M)$$ Lip 0 ( M ) consisting of strongly norm-attaining Lipschitz functionals, for $$n\in {\mathbb {N}}$$ n ∈ N . We show that this is always the case for infinite metric spaces, providing a definitive answer to the question. We also study the possible sizes of such infinite-dimensional closed linear subspaces Y, as well as the inverse question, that is, the possible sizes for a metric space M in order to such a subspace Y exist. We also show that if the metric space M is $$\sigma $$ σ -precompact, then the aforementioned subspaces Y need to be always separable and isomorphically polyhedral, and we show that for spaces containing [0, 1] isometrically, they can be infinite-dimensional.

On isometric embeddings into the set of strongly norm-attaining Lipschitz functions

“…In the past few years, this topic has been intensively studied. We send the reader to [1,2,3,4,5,6,7,11,12,14,15,17,18,19] and the references therein.…”

“…We consider the subset SNA(M) of Lip 0 (M) of all strongly norm-attaining Lipschitz functions on M. In 2016, Marek Cúth, Michal Doucha, and Przemysław Wojtaszczyk [8] proved that ℓ ∞ (and hence c 0 ) embeds isomorphically in Lip 0 (M) for any infinite metric space M. One year later, this result was improved by Marek Cúth and Michal Johanis and it is known now that ℓ ∞ (and hence c 0 ) embeds isometrically in Lip 0 (M) [9]. Motivated by the papers [1,13,19], we turn our attention to the study of the analogous problems for the subset SNA(M). Drastically different from the classical norm-attaining theory, where Martin Rmoutil [21] proved that the set of all norm-attaining functionals needs not contain 2-dimensional spaces, Antonio Avilés, Gonzalo Martínez Cervantes, Abraham Rueda Zoca, and Pedro Tradacete [1] provided a beautiful and interesting construction, and showed that SNA(M) contains an isomorphic copy of c 0 for every infinite metric space M (answering [19,Question 2]).…”

“…Motivated by the papers [1,13,19], we turn our attention to the study of the analogous problems for the subset SNA(M). Drastically different from the classical norm-attaining theory, where Martin Rmoutil [21] proved that the set of all norm-attaining functionals needs not contain 2-dimensional spaces, Antonio Avilés, Gonzalo Martínez Cervantes, Abraham Rueda Zoca, and Pedro Tradacete [1] provided a beautiful and interesting construction, and showed that SNA(M) contains an isomorphic copy of c 0 for every infinite metric space M (answering [19,Question 2]). At the very ending of that paper, the authors wondered whether an isometric version of this result holds true (see [1,Remark 3.6]).…”